This smells like cryptography or similar task. If you just wanna store your value, put your integer factorization into an array. It’s that simple. Then the entire array together represents a value, or at least a way to calculate it, you know. You don’t need to calculate the product, just store its factors. Use (a sorted array of) primes if you want a unique way to store a product.
Oh, I did need to calculate the product, and I used the library gmp (you'd suggested me here) for calculating that.
Actually, I wanted to see if the equation |3^x - 2^y| = 1, had any natural solutions, besides the three obvious ones:
3^1 - 2^1 = 1.
2^2 - 3^1 = 1.
3^2 - 2^3 = 1.
My conjecture is that this equation has no other natural solutions. Thanks to the library you had suggested me here, my conjecture turned out to be true - for all natural exponentials not larger than 10,000. However. for larger natural exponentials, the calculation becomes too slow, so I didn't check out any larger natural exponentials.
Thank you so much , for being the first one to suggest me a very useful library - by which I could verify my conjecture for very big natural numbers. I still wonder, though, if my conjecture is provable for all natural numbers.
Thanks also to the other members for the other suggestions.